Least-squares inverse filtering

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A well-behaved input wavelet, such as (1, -  1/2) as opposed to (-  1/2, 1), has a z-transform whose inverse can be represented by a convergent series. Then the inverse filtering described above yields a good approximation to a zero-lag spike output (1, 0, 0). Can we do even better than that?

Formulate the following problem: Given the input wavelet (1, -  1/2), find a two-term filter (a, b) such that the error between the actual output and the desired output (1, 0, 0) is minimum in the least-squares sense.

Compute the actual output by convolving the filter (a, b) with the input wavelet (1, -  1/2) (Table 2-7). The cumulative energy of the error L is defined as the sum of the squares of the differences between the coefficients of the actual and desired outputs:


(14)

The task is to find coefficients (a, b) so that L takes its minimum value. This requires variation of L with respect to the coefficients (a, b) to vanish (Section B.5). By simplifying equation (14), taking the partial derivatives of quantity L with respect to a and b, and setting the results to zero, we get


(15a)

and


(15b)

We have two equations and two unknowns; namely, the filter coefficients (a, b). The so-called normal set of equations (15a) and (15b) can be put into the following convenient matrix form


(16)

By solving for the filter coefficients, we obtain (a, b): (0.95, 0.38). Design and application of this least-squares inverse filter are summarized in Table 2-7.

To quantify the spikiness of this result and compare it with the result from the inverse filter in Table 2-3, compute the energy of the errors made in both (Table 2-8). Note that the least-squares filter yields less error when trying to convert the input wavelet (1, -  1/2) to a spike at zero lag (1, 0, 0).

Table 2-3. Design and application of the truncated inverse filter (1, 1/2) with the input wavelet (1, -  1/2).
Filter Design
Input Wavelet
The z-Transform
The Inverse
The Inverse Filter
Filter Application
Truncated Inverse Filter
Input Wavelet
Actual Output
Desired Output (1, 0, 0)
Convolution Table:
1 -  1/2 Output
1/2 1 1
1/2 1 0
1/2 1 -  1/4

We now examine the performance of the least-squares filter with the input wavelet (-  1/2, 1). Note that the inverse filter produced unstable results for this wavelet (Table 2-5). We want to find a two-term filter (a, b) that, when convolved with the input wavelet (-  1/2, 1), yields an estimate of the desired spike output (1, 0, 0) (Table 2-9). As before, the least-squares error between the actual output and the desired output should be minimal.

Table 2-5. Design and application of the truncated inverse filter (−2, −4) with input wavelet (-  1/2, 1).
Filter Design
Input Wavelet
The z-Transform
The Inverse
The Inverse Filter f(t): (−2, −4, −8, …)
Filter Application
Truncated Inverse Filter (−2, −4)
Input Wavelet
Actual Output (1, 0, −4)
Desired Output (1, 0, 0)
Convolution Table:
-  1/2 1 Output
−4 −2 1
−4 −2 0
−4 −2 −4

The cumulative energy of the error is given by


(17)
Table 2-7. Design and application of a two-term least-squares inverse filter (a, b).
Filter Design
Convolution of the filter (a, b) with input wavelet (1, -  1/2):
1 -  1/2 Actual Output Desired Output
b a a 1
b a b − a/2 0
b a b/2 0
Filter Application
Least-Squares Filter (0.95, 0.38)
Input Wavelet (1, −0.5)
Actual Output (0.95, −0.09, −0.19)
Desired Output (1, 0, 0)
Table 2-8. Error in two-term inverse and least-squares filtering.
Input:
Desired Output: (1, 0, 0)
Actual Output Error Energy
Inverse Filter (1, 0, −0.25) 0.063
Least-Squares Filter (0.95, −0.09, −0.19) 0.048
Table 2-9. Design and application of a two-term least-squares inverse filter (a, b).
Filter Design
Convolution of the filter (a, b) with input wavelet :
-  1/2 1 Actual Output Desired Output
b a a/2 1
b a b/2 + a 0
b a b 0
Filter Application
Least-Squares Filter (−0.95, −0.19)
Input Wavelet (−0.5, 1)
Actual Output (0.24, −0.38, −0.19)
Desired Output (1, 0, 0)
Table 2-10. Error in two-term inverse and least-squares filtering.
Input:
Desired Output: (1, 0, 0)
Actual Output Error Energy
Inverse Filter (1, 0, −4) 16
Least-Squares Filter (0.24, −0.38, −0.19) 0.792

By simplifying equation (17), taking the partial derivatives of quantity L with respect to a and b, and setting the results to zero, we obtain


(18a)

and


(18b)

Combine equations (18a,18b) into a matrix form


(19)

By solving for the filter coefficients, we obtain (a, b): (−0.95, −0.19). The design and application of this filter are summarized in Table 2-9.

Table 2-10 shows the results from the inverse filter and least-squares filter quantified. The error made by the least-squares filter is, again, much less than the error made by the truncated inverse filter. However, both filters yield larger errors for input wavelet (-  1/2, 1) (Table 2-10) as compared to errors for wavelet (1, -  1/2) (Table 2-8). The reason for this is discussed next.

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Least-squares inverse filtering
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